We study the connection between intersection properties of balls and the existence of large faces of the unit ball in Banach spaces. Hanner’s result that a real space has the 3.2 intersection property if an only if disjoint faces of the unit ball are contained in parallel hyperplanes is extended to infinite dimensional spaces. It is shown that the space of compact operators from a space X to a space Y has the 3.2 intersection property if and only if X and Y have the 3.2 intersection property and either X or Y* is isometric to an L1(μ)-space.